Can Eigenvalues Define Solutions for PDE Boundary Problems?

[Pyroadept]

Homework Statement

Let λ_n denote the nth eigenvalue for the problem:

-Δu = λu in A, u=0 on ∂A (*)

which is obtained by minimizing the Rayleigh quotient over all non-zero functions that vanish on ∂A and are orthogonal to the first n-1 eigenfunctions.

(i) Show that (*) has no other eigenvalues

(ii) Show that one can find a non-negative eigenfunction for the first eigenvalue of (*). Such an eigenfunction is also known as a ground state.
Hint: If w is an eigenfunction that changes sign, then α=max(w,0) and β=min(w,0) are nonzero but of one sign; relate their Rayleigh quotients with that of their sum α+β=w


Now let some real number 'a' be fixed and f(x) a given smooth function. Consider the problem:

-Δu = au + f in A, u=0 on ∂A (**)

(iii) Show (**) has no solution when a = λ_k is an eigenvalue of (*) and f is not orthogonal to the corresponding eigenfunction u_k.

(iv) Let f(x) be a given smooth function and suppose a is not an eigenvalue of (*). Find the unique solution of the problem (**). Hint: the solution is a linear combination of the eigenfunctions u_k by completeness, so you need only determine the coefficients.

Homework Equations

The Attempt at a Solution

Hi guys,

Thanks for looking at my problem, I am a bit stuck on this particular one! We didn't do much about eigenvalues (with regard to pde's) in class and I haven't been able to find any books that give examples other than the basic theorems. Here's what I've done so far:

(i) Use proof by contradiction. Let λ* be some other eigenvalue, ie λ_n < λ* < λ_(n+1)

thus -Δu = λ*u

for u a corresponding eigenfunction.

The Rayleigh quotient is:

R(u) = ( ∫|∇u|^2 ) / ( ∫∇u^2 )

Suppose R(u) attains a minimum over the set:

X_n = { u ∊ C^2(A): u=0 on ∂A, u ≢0 and u⊥u_1, ..,u_(n-1) }
Then any function that minimizes R(u) over X_n is an eigenfunction u_n, with eigenvalue λ_n,

where λ_n = min R(u), for u ∊ X_n.


Now we want to show that λ* is not of this form i.e. minimizes R(u) but not over the set as outlined above.
But if λ* is indeed an eigenvalue, and minimizes R(u), with corresponding eigenfunction u* say, then there will be some u ∊ {u_1, u_2, ... , u_n ... }, u_k, say, such that u_k ⊥ u*.
But then u_k will be orthogonal to u_1, u_2, ..., u_(k-1), and then so will u*.

But u* cannot be a member of this set.
And this is a contradiction.

Thus all λ are indeed of the required form.


I don’t know if this is right (I think I may have left out something, it just seems a bit straightforward...) I would be really grateful if you could tell me if it is?

(ii)
By definition of the Rayleigh quotient,

R(α) = ( ∫|∇ α |^2 ) / ( ∫∇ α ^2 )

and

R(β) = ( ∫|∇ β |^2 ) / ( ∫∇ β ^2 )

and

R(α +β) = ( ∫|∇ α |^2 + 2∫ ∇ α .∇ β + ∫|∇ β |^2 ) / ∫( α^2 + 2αβ + β^2)

But then I’m stuck as to how to relate them, is it a case of minimizing R(α +β)?.



(iii)
If a= λ_k is an eigenvalue of (*), then:

-Δu = au

But -Δu = au+f

So then au = au+f, unless f orthogonal (in which case it vanishes).
So this gives us a contradiction otherwise.

I know that seems really brief, but it is making sense in my head, I think I’m just not articulating it properly. Please could someone give me a point in the right direction?


(iv)
I actually have no idea how to go about solving this one. How do I go about setting up the equation and solving the coefficients?


Thank you so much for looking at this and I would really appreciate any hints or pointers you could give me. Thanks again.

 

[mighty2000]

Dear ,

Thank you for posting your problem on the forum. I will do my best to help you solve it.

For part (i), your proof by contradiction is correct. You have shown that any other eigenvalue must also be of the form λ_n, thus proving that there are no other eigenvalues.

For part (ii), you are on the right track. You need to minimize R(α+β) and show that it is equal to R(u). You can do this by using the fact that α and β are both non-negative, thus minimizing their sum will also minimize the Rayleigh quotient.

For part (iii), you are correct in saying that f must be orthogonal to u_k in order for the equation to have a solution. This is because if f is not orthogonal, it will not vanish and thus cannot be equal to 0 on the boundary.

For part (iv), you can use the fact that any smooth function can be written as a linear combination of the eigenfunctions u_k. Then, by using the orthogonality of the eigenfunctions, you can solve for the coefficients and find the unique solution to the problem.

I hope this helps you solve the problem. If you need any further clarification or assistance, please do not hesitate to ask. Good luck with your studies!